Equation of an Elliptical Orbit for a Moon around a Planet

In summary, The conversation is about determining an equation for the elliptical orbit of Oberon around Uranus using data from four nights. The main issue is figuring out how to apply the RA and Dec coordinates to the equation of an ellipse. The eccentricity of the ellipse fitted to the data is much higher than expected, potentially due to the significant change in angular separation between Uranus and Oberon. The inclination of Oberon's orbit is also mentioned as a possible factor. The person is seeking help in finding where they went wrong in their calculations.
  • #1
farore
2
0
I am kind of new to astronomy and I have a few questions..

I need to be able to determine an equation expressing the elliptical orbit of Oberon around Uranus. I have four nights of data showing Oberon in different positions relative to Uranus. I am able to calculate the RA and Dec for both Uranus and Oberon from these frames (standard stars are on frame).

I just need to be able to determine the semi-major axis of the orbit so that I can determine the eccentricity. What I am most stuck on is how to apply my RA and Dec coordinates to an equation of an ellipse.

I saw a similar discussion in this thread (astrometry of elliptical orbits) but it was for asteroids and at first glance the analysis involved seemed more complicated than what I was going for.

Thanks for any help.
 
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  • #2
Nevermind about the first bit, I figured out that I needed to express the change in RA versus the change in Dec of Oberon with respect to Uranus to get an ellipse.

However, with my data points the eccentricity of the ellipse I fit to the data is on the order of .9 when it should be around .008 or so.

The angular separation between Uranus and its moon goes from about 20 arc seconds to 40 arc seconds, which apparently is significant when determining the eccentricity of the ellipse. Am I missing something? The inclination of Oberon with respect to Uranus' equator is about .012 degrees so I didn't think I would have to rotate it into the plane since that seems like a fairly negligible quantity (and I don't think it would make a .008 eccentricity turn into a .9 eccentricity).

Any help on determining where I went wrong would be much appreciated :).
 

Related to Equation of an Elliptical Orbit for a Moon around a Planet

1. What is the equation of an elliptical orbit for a moon around a planet?

The equation for an elliptical orbit is a mathematical representation of the path that a moon takes while orbiting a planet. It is given by r = a(1 - ecosθ), where r is the distance between the moon and the planet, a is the semi-major axis of the ellipse, e is the eccentricity of the ellipse, and θ is the angle between the moon and the periapsis (closest approach to the planet).

2. How is the equation of an elliptical orbit derived?

The equation of an elliptical orbit is derived from Kepler's laws of planetary motion. The second law states that a line connecting a planet to the sun sweeps out equal areas in equal intervals of time. This translates to a constant rate of change of the angle θ with respect to time, which is used in the equation of an elliptical orbit.

3. What is the significance of the semi-major axis and eccentricity in the equation of an elliptical orbit?

The semi-major axis, denoted by a, is half the length of the longest diameter of the ellipse. It determines the size of the orbit, with larger values indicating a larger orbit. The eccentricity, denoted by e, measures the degree of deviation from a perfect circle. A value of 0 indicates a circular orbit, while values closer to 1 indicate a more elongated elliptical orbit.

4. Can the equation of an elliptical orbit be used for any moon-planet system?

Yes, the equation of an elliptical orbit can be used for any moon-planet system, as long as the moon is orbiting the planet in an elliptical path. This includes the Earth-Moon system, where the moon's orbit around the Earth is an ellipse with a semi-major axis of about 384,400 km and an eccentricity of 0.0549.

5. How does the equation of an elliptical orbit change if the moon's orbit is highly eccentric?

If the moon's orbit is highly eccentric, the semi-major axis and eccentricity values in the equation will be larger, resulting in a more elongated ellipse. This means that the moon will have a larger distance from the planet at its farthest point (apoapsis) and a closer distance at its closest point (periapsis). The angle θ will also change more rapidly as the moon moves through its orbit.

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